Fuel cell system has an anode chamber and a cathode chamber separated from one another by a proton-conducting membrane. A cathode supply line supplies gas containing oxygen to the cathode chamber, and an anode supply line supplies a liquid coolant/fuel mixture to the anode chamber. The anode chamber is located in an anode circuit that includes a gas separator and a pump, and with a cooler and a water separator being located in the separated gas flow from the gas separator in the flow direction. According to the invention, the anode circuit has an operating pressure of at least 2.5 bars. The cooler is also cooled by a medium at ambient temperature.